A manufacturer constructs open boxes from sheets of cardboard that are 6 inches square by cutting small squares from the corners and folding up the sides. The Research and Development Department asks you to determine the size of the square that produces a box of greatest volume. Proceed as follows. Let x be the length of a side of the square to be cut and V be the volume of the resulting box. Show that V = x(6-2x)^2. I don't understand how I'm suppose to show what they are asking.

Question

A manufacturer constructs open boxes from sheets of cardboard that are 6 inches square by cutting small squares from the corners and folding up the sides. The Research and Development Department asks you to determine the size of the square that produces a box of greatest volume. Proceed as follows. Let x be the length of a side of the square to be cut and V be the volume of the resulting box. Show that V = x(6-2x)^2.  I don't understand how I'm suppose to show what they are asking.

Reiny · Answer

make a sketch of a 6 by 6 square.
Draw squares at each corner of x by x
(You could actually cut them out with scissors)
This would allow you to fold up the sides to form a box
Wouldn't each side of the base remaining be 6 - 2x, since x units were taken away at each end ?
Wouldn't the height of the box be x ?

volume = base x height
= (6-2x)(6-2x(x)
= x(6-2x)^2

Isaac · Answer

I just wasn't sure if the answer was simply a sketch, 6 - 2x each side with the cut-outs. Thanks!!